Onsager reaction field model

The simplest SCRF model is the Onsager reaction field model. In this method, the solute occupies a fixed spherical cavity of radius Oq within the solvent field. A dipole in the molecule will induce a dipole in the medium, and the electric field applied by the solvent dipole will in turn interact with the molecular dipole, leading to net stabilization. 

The salient features of quantum formulation of Onsager reaction field model (dipole model) is described here. In this method, the reaction field is treated as perturbation to the Hamiltonian of the isolated molecule. If H0 is the Hamiltonian of the isolated molecule and HR[ is the reaction field [21], the Hamiltonian of the whole system (Hlol) is represented as... 

As a first approximation, solvent effects can be described by models where the solvent is represented by a dielectric continuum, e.g., the Onsager reaction-field model. 

In practice, empirical or semi-empirical interaction potentials are used. These potential energy functions are often parameterized as pairwise additive atom-atom interactions, i.e., Upj(ri,r2,..., r/v) = JT. u ri j), where the sum runs over all atom-atom distances. An all-atom model usually requires a substantial amount of computation. This may be reduced by estimating the electronic energy via a continuum solvation model like the Onsager reaction-field model, discussed in Section 9.1. 

The Cl relaxed density approach [18] should give a more accurate evaluation of the reaction field, but because of its more involved computational character it has been rarely applied in Cl solvation models. The only notably exception is the Cl methods proposed by Wiberg at al. in 1991 [19] within the framework of the Onsager reaction field model. In their approach, the electric dipole moment of the solute determining the solvent reaction field is not given by an expectation value but instead it is computed as a derivative of the solute energy with respect to a uniform electric field. 

A more general framework to treat local field effects in linear and nonlinear optical processes in solution has been pioneered, among others [45], by Wortmann and Bishop [46] using a classical Onsager reaction field model (see the contribution by the Cammi and Mennucci for more details). Such a model has not been extended to treat vibrational spectra. 

Probably the simplest quantum-mechanical operators that include interaction with a continuum are the Born and Onsager reaction field models. In the case of neutral solutes, the model could be express as... 

This is a generalization of the Onsager reaction field model for a point dipole inside a spherical cavity. For charged solutes, one should also include an ionic Born term, derived by... 

The experimental data reported in the Table for gas phase have been extracted from measurements in dioxane solution by applying the Onsager reaction field model to eliminate the solvent effect [37], By contrast, the cyclohexane experimental dipole moments have been obtained from those reported in Ref. [37] re-including the proper reaction field factors. Once recalled these facts, we note that the observed solvent-induced changes on both ground and excited state dipole moments are quantitatively reproduced by the calculations. 

The first semiempirical studies of the potential functions of silatranes revealed that their Si-<—N bond could be very easily deformed by crystal packing and solvation forces . Direct MINDO/3 and MNDO calculations demonstrate a significant shortening of the equilibrium, gas phase Si- —N distance due to the contribution of the solute-solvent interaction energy (within the Onsager reaction field model ) to the total enragy of 1-methyl- and 1-fluorosilatranes . 

Bishop [207] using a classical Onsager reaction field model, but such a model has not been extended to vibrational spectra. 

S (and of the interaction between M and S) developed beyond the original Onsager reaction field model. 

Onsager s reaction field model in its original fonn offers a description of major aspects of equilibrium solvation effects on reaction rates in solution that includes the basic physical ideas, but the inlierent simplifications seriously limit its practical use for quantitative predictions. It smce has been extended along several lines, some of which are briefly sunnnarized in the next section. 

In the reaction field model (Onsager, 1936), a solute molecule is considered as a polarizable point dipole located in a spherical or ellipsoidal cavity in the solvent. The solvent itself is considered as an isotropic and homogeneous dielectric continuum. The local field E at the location of the solute molecule is represented by (78) as a superposition of a cavity field E and a reaction field (Boettcher, 1973). 

Local ordering effects have long been recognised experimentally in measurements of dipole moments of polar solutes in non-polar solvents, where the value obtained on the basis of the simple model differs from the value obtained for the pure solute in the gas phase, even when the results are extrapolated to infinite dilution. This so-called solvent effect is due to the Onsager reaction field. If there is no strong local ordering, Onsager s formula (2.52) is valid and the apparent solution moment is related to the isolated molecule or gas moment by... 

Born—Kirkwood—Onsager Reaction Field The theory underlying the implementation of the BKO model at the semiempirical level is no different from that presented in Equations [22] and [23], although the approximations inherent to various levels of semiempirical theory make certain technicalities of the... 

In the classical reaction-field model, the solute molecule is considered embedded in a cavity inside a homogeneous dielectric medium. From the Onsager theory (Onsager, 1936) the electronic reaction field at the center of the solute molecule is given by... 

In addition, the molecules properties are changed due to the interaction with the surrounding medium. Several computational schemes have been proposed to address these effects. Tliey are essentially based on the extension of the Onsager reaction field cavity model and give effective hyperpolarizabilities, i.e. molecular hyperpolarizabilities induced by the external fields that include the modifications due to the surrounding molecules as well as local (cavity) field effects [40 2]. These condensed-phase effects have, however, not yet been included in the SFG hyperpolarizability calculations, which are therefore strictly gas-phase calculations. 

While this result confirmed the feasibility of the general approach, it did not precipitate wider exploration of dielectric medium effects. Recently, however, Wiberg et al. have incorporated the Onsager self-consistent reaction-field model into ab initio MO theory in an implementation which provides analytical gradients and second derivatives. The model considers just the dipole of the solute molecules and a spherical cavity whose radius is chosen for a given solute molecule from the molecular volume estimated at the 0.001 eB electron-density contour (B is the Bohr radius), plus an empirical constant 0.5 A to account for the nearest approach of solvent molecules [164]. Cieplak and Wiberg have used this model to probe solvent effects on the transition states for nucleophilic additions to substituted acetaldehydes [165]. For each... 

In view of the approximations inherent in the derivation of the reaction field theory, it is not surprising that some instances are known in which a non-linear relationship exists between the solvent shift and dielectric constant in polar solvents. As pointed out by Buckingham, the reaction field model is only valid for a solute that reacts in no way with the solvent or with other solute molecules but simply presents a continuum of certain dielectric properties. Protons are normally on the surface of the molecule and are therefore exposed to direct contact with the surrounding molecules, so that the Onsager model is a poor approximation of the actual reaction field acting on a molecule. 

Wortmann and Bishop14 have successfully extended Onsager s148 reaction field model to non-linear optical properties in condensed media,147 which is applicable in any order of dynamic hyperpolarizabilities. 

The simplest continuum model is based on die classical Onsager reaction field theory assuming the spherical or ellipsoidal form of cavities for die solute molecules in dielectric... 

---